I haven't really been able to find an answer to this question. I've only noticed that most instruments during the baroque period were tuned a semitone lower (A=415 Hz I believe). So shouldn't it make sense to play baroque pieces in a key which is a semitone lower than the original or tune the instrument down by a semitone? (Particularly interested in pieces written by Bach)
Musical pitch varied widely from place to place during the Baroque Era. It was generally lower than it is today, but there were exceptions and sometimes it could be even higher. Modern measurements of baroque instruments show that the pitch would mostly have varied between A=400 and A=450 Hz (in extreme cases from about A=380 to A=480 Hz) .
Modern ensembles that play baroque music wanted to use a lower pitch than the modern standard, and they settled on A=415 Hz, so most modern reproductions of baroque instruments are built to that pitch. A=415 has the advantage that it is very close to a semitone below A=440 and it's possible to play together with modern instruments transposed down a semitone. Keyboard instruments can easily be switched from one pitch to the other by shifting the keyboard relative to the strings. Other than convenience, there is nothing special about the choice of a semitone.
For performances on modern instruments, the pitch options depend on the instruments involved. Woodwind instruments in particular can't be retuned down, and transposing a piece down a semitone can turn an easy key into an almost impossible one. Tuning string instruments down can have the problem that the strings get too loose. Electronic keyboards, of course, can be tuned to any pitch.
With the exception of specialist baroque ensembles, the vast majority of performances of baroque music are at modern pitch, and that is what people are used to hearing. There's nothing to compel a performer to use a lower pitch, but there's nothing to stop them doing it where it's possible. The majority of listeners would probably not even notice a difference. It's up to the performer to decide what works and what doesn't.
It would make sense to play them on authentic instruments, at the original pitch, and many performance groups make a speciality of doing this. But there's more to authenticity than just the pitch. I don't see a lot of point in tuning modern instruments down. Maybe if voices are involved.
There's also the question of styles of performance. Even within the era of recorded music, there have been changes. We can now compare styles of performing the same 'classical' pieces 100 years ago and today. And it's very instructive to do so! I think we can safely assume there were similar changes in the preceding 100 years, and the 100 before that. And, although we have been left clues, a lot of our ideas of what was 'authentic' are supposition.
It really depends on the instruments you have available. Bach is not an equal temperament composer, and the instruments of his period are not equal temperament instruments. Even more modern versions of flutes, lutes, string instruments, brass and so on have dominant scales and are built for a particular tuning. Mapping modern instruments to a different key or even "merely" retuning will more likely than not change more of the character than the different frequency as such does.
Singers are a different matter: in fact they are usually built quite larger than in baroque times and expedients like boy sopranos have fallen out of use (let alone baroque castrati). So if you have an orchestra with period instruments in period tuning, tuning down the singers is usually considered the least of problems since more often than not there is a dearth of sopranos and particularly tenors anyway, and few singers have absolute pitch.